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On a Van Kampen theorem for Hawaiian groups
Topology and its Applications 241 (2018) 252–262
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Topology and its Applications www.elsevier.com/locate/topol
On a Van Kampen theorem for Hawaiian groups Ameneh Babaee, Behrooz Mashayekhy ∗ , Hanieh Mirebrahimi, Hamid Torabi Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhad, Iran
a r t i c l e
i n f o
Article history: Received 13 July 2017 Received in revised form 17 March 2018 Accepted 8 April 2018 Available online 10 April 2018 MSC: 55Q05 55Q20 55P65 55Q52
a b s t r a c t The paper is devoted to study the nth Hawaiian group Hn , n ≥ 1, of the wedge sum of two spaces (X, x∗ ) = (X1 , x1 ) ∨(X2 , x2 ). We are going to give some versions of the van Kampen theorem for Hawaiian groups of the wedge sum of spaces. First, among some results on Hawaiian groups of semilocally strongly contractible spaces, we present a structure for the nth Hawaiian group of the wedge sum of CW-complexes. Second, we give more informative structures for the nth Hawaiian group of the wedge sum X, when X is semilocally n-simply connected at x∗ . Finally, as a consequence, we study Hawaiian groups of Griffiths spaces for all dimensions n ≥ 1 to give some information about their structure at any points. © 2018 Elsevier B.V. All rights reserved.
Keywords: Hawaiian group Hawaiian earring Van Kampen theorem Griffiths space
1. Introduction and motivation In 2006, U.H. Karimov and D. Repovš [6] defined the nth Hawaiian group as a covariant functor Hn : hT op∗ → Groups from the pointed homotopy category, hT op∗ , to the category of all groups, Groups, for n ≥ 1. For any pointed space (X, x0 ), the nth Hawaiian group Hn (X, x0 ) was defined to be the set of all pointed homotopy classes [f ], where f : (HEn , θ) → (X, x0 ) is a pointed map. Here, HEn denotes the n-dimensional Hawaiian earring, the union of n-spheres Snk in Rn+1 with centre (1/k, 0, . . . , 0) and radius 1/k, and the origin θ is considered as the base point. The operation of the nth Hawaiian group arises component-wisely from the operation of the nth homotopy group. Thus the following map ϕ : Hn (X, x0 ) → πn (X, x0 ), (I) N
* Corresponding author. E-mail addresses: [email protected] (A. Babaee), [email protected] (B. Mashayekhy), [email protected] (H. Mirebrahimi), [email protected] (H. Torabi). https://doi.org/10.1016/j.topol.2018.04.005 0166-8641/© 2018 Elsevier B.V. All rights reserved.
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defined by ϕ([f ]) = ([f |Sn1 ], [f |Sn2 ], ...) is a homomorphism, for all n ∈ N. Note that in general, the homomorphism ϕ is not injective nor surjective. Let C(HE1 ) be the cone over 1-dimensional Hawaiian earring. Since C(HE1 ) is simply connected and H1 (C(HE1 ), θ) is not trivial [6], ϕ can not be injective. Moreover, W for every locally n-simply connected first countable space X, im(ϕ) = N πn (X, x0 ) [6, Theorem 1] which is a proper subgroup of N πn (X, x0 ) whenever πn (X, x0 ) is nontrivial. Here, we consider the weak direct W product N πn (X, x0 ) as a subgroup of Hn (X, x0 ), because it is proved that for every pointed space (X, x0 ), W N πn (X, x0 ) can be embedded as a normal subgroup in Hn (X, x0 ) (see [1, Lemma 2.4] and [1, Theorem 2.13]). Although the nth Hawaiian group functor is a pointed homotopy invariant functor on the category of all pointed topological spaces, it is not freely homotopy invariant. Because unlike other homotopy invariant functors, Hawaiian groups of contractible spaces are not necessarily trivial. Karimov and Repovš [6] gave a contractible space, the cone over HE1 , with nontrivial 1st Hawaiian group at some points (consisting of the points at which CHE1 is not locally 1-simply connected), but with trivial homotopy, homology and cohomology groups. More precisely, it can be shown that H1 (C(HE1 ), θ) is uncountable, using [6, Theorem 2]: “if X has a countable local basis at x0 , then countability of nth Hawaiian group Hn (X, x0 ) implies locally n-simply connectedness of X at x0 .” Furthermore, a converse of the above statement can be found in [1, Corollaries 2.16 and 2.17]: “let X have a countable local basis at x0 . Then Hn (CX, x ˜) is trivial if and only if X is locally n-simply connected at x0 and it is uncountable otherwise”. In addition, unlike homotopy groups, Hawaiian groups of pointed space (X, x0 ) depend on the behaviour of X at x0 , and then their structures depend on the choice of base point. In this regard, there exist some examples of path connected spaces with non-isomorphic Hawaiian groups at different points, such as the n-dimensional Hawaiian earring, where n ≥ 2 (see [1, Corollary 2.11]). Despite the above different behaviours between Hawaiian groups and homotopy groups, they have some similar behaviours. For instance, it was proved that similar to the nth homotopy group, the nth Hawaiian group of any pointed space is abelian, for all n ≥ 2 [1, Theorem 2.3]. Also, the Hawaiian groups preserve products in the category hT op∗ [1, theorem 2.12]. In this paper, we investigate the Hawaiian groups of the coproduct in the category hT op∗ which is the wedge sum of a given family of pointed spaces. In fact, we are going to give some versions of the van Kampen theorem for Hawaiian groups of the wedge sum of spaces. In Section 2, among some results on Hawaiian groups of semilocally strongly contractible spaces, we intend to present a structure for the nth Hawaiian group of the wedge sum of CW-complexes. A given space X is semilocally strongly contractible at x0 if the inclusion i : U → X is nullhomotopic in X relative to {x0 }, for some open neighbourhood U of x0 (see [4]). In Section 3, we present the nth Hawaiian group of the wedge sum (X, x∗) = (X1 , x1 ) ∨ (X2 , x2 ) as the direct product of two its subgroups that are more perceptible, when X is semilocally n-simply connected at x∗ . Also, we prove that the Hawaiian group of a pointed space equals the Hawaiian group of every neighbourhood of the base point if all n-loops are small. An n-loop α : (Sn , 1) → (X, x) is called small, if for each neighbourhood U of x, α has a homotopic representative in U (see [7,10]). In Section 4, we investigate the Hawaiian groups of the nth Griffiths space, introduced in [4, Corollary 1.4], by generalizing the well-known Griffiths space, as the wedge sum of two copies of the cone over the n-dimensional Hawaiian earring. For the sake of clarity, we call the well known Griffiths space as the 1st Griffiths space. Then, using results of Sections 2 and 3, we intend to give some information about the structure of Hawaiian groups of Griffiths spaces at any points. In this paper, all homotopies are relative to the base point, unless stated otherwise.
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Fig. 1. The 1st Griffiths space.
2. Hawaiian groups of semilocally strongly contractible spaces In this section, we investigate Hawaiian groups of wedge sum of pointed spaces which are semilocally strongly contractible at the base points. The property semilocally strongly contractibility was defined by Eda and Kawamura [4]. Definition 2.1. A space X is called semilocally strongly contractible at x0 if there exists some open neighbourhood U of x0 such that the inclusion map i : U → X is nullhomotopic in X relative to {x0 }, in other words, i : (U, x0 ) → (X, x0 ) is nullhomotopic. First, we compare semilocally strongly contractible property with some familiar properties, such as locally contractible, locally strongly contractible and semilocally contractible properties. Recall 2.2. Let (X, x) be a pointed space, then X is called locally contractible at x if for each open neighbourhood U of x in X, there exists some open neighbourhood V of x contained in U such that the inclusion V → U is freely nullhomotopic. We say that X is locally strongly contractible at x if for each open neighbourhood U of x in X, there exists some open neighbourhood V of x contained in U such that the inclusion V → U is nullhomotopic relative to {x}, or briefly, (V, x) → (U, x) is nullhomotopic. Moreover, X is called semilocally contractible at x, if there exists some open neighbourhood U of x such that the inclusion i : U → X is freely nullhomotopic in X. One can see that there are some relations between these properties. Locally strongly contractible property implies locally contractibility, and locally contractible property implies semilocally contractibility, but the converse statements do not necessarily hold. Moreover, locally strongly contractible property implies semilocally strongly contractibility, and semilocally strongly contractible implies semilocally contractibility, but converse statements do not necessarily hold. The following example shows that a given space can behave differently at different points. Example 2.3. The 1st Griffiths space G1 is locally strongly contractible at two vertices v and v . Therefore, it is locally contractible, semilocally strongly contractible and semilocally contractible at two vertices v and v . Moreover, G1 is semilocally contractible at all points except at the common point x∗ . Also, G1 is semilocally contractible at any point of A and A , but not semilocally strongly contractible, where A and A are open
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intervals from the common point x∗ to the corresponding vertices. Finally, G1 is neither semilocally strongly contractible, semilocally contractible nor even semilocally 1-simply connected at x∗ (see [5]). The 1-dimensional Hawaiian earring HE1 is not semilocally contractible at the origin. However, it is locally strongly contractible at other points. There exist many spaces which are semilocally contractible at each point, but not strongly at some points. For example, consider CΔ as the cone over the Seirpeinski gasket. It is semilocally contractible at any points, but it is semilocally strongly contractible just at the vertex. In the following theorem, we prove that Hawaiian group is generated by two subgroups depend on local behaviour at the given point. It helps us to investigate Hawaiian groups of spaces having specific local properties. Theorem 2.4. Let (X, x0 ) be a pointed space and n ≥ 1. Then Hn (X, x0 ) = j∗ Hn (U, x0 ) each open neighbourhood U with x0 ∈ U ⊆ X and j : U → X as the inclusion map.
W N
πn (X, x0 ), for
Proof. Let U be any open neighbourhood of x0 , and j : U → X be the inclusion map. Also, let f : (HEn , θ) → (X, x0 ) be a pointed map. Since U is open in X, there exists K ∈ N such that if k ≥ K, then im(f |k≥K Snk ) ⊆ U . We can define f , f : (HEn , θ) → (X, x0 ) by f |k
C|k≥K Snk , f |k
at x0 . Recall that for any classes, [f ] ∗ [g] is obtained by (f ∗ g)|Snk = f |Snk ∗ g|Snk . Hence, [f ] = [f ][f ] = [f ][f ]. W Moreover, [f ] is an element of N πn (X, x∗ ), and also, [f ] is an element of j∗ Hn (U, x0 ). Therefore, Hn (X, x∗ ) W W is generated by j∗ Hn (U, x0 ) ∪ N πn (X, x0 ). Since N πn (X, x0 ) is a normal subgroup of Hn (X, x0 ) (see the proof of [1, Theorem 2.13]), the equality holds. 2 The following lemma describes the Hawaiian group of the wedge sum of two spaces by its subgroups. This construction helps us to obtain next isomorphisms. Let {Gi }i∈I be a family of indexed groups and ei W be the identity element of Gi . Then by i∈I Gi , we mean the weak direct product of the family {Gi }i∈I consisting of all elements {gi }i∈I of i∈I Gi such that gi = ei , for all i ∈ I except a finite number. Corollary 2.5. Let (X1 , x1 ) and (X2 , x2 ) be two pointed spaces, (X, x∗ ) = (X1 , x1 ) ∨ (X2 , x2 ), and n ≥ 1. (i) If U1 and U2 are two open neighbourhoods of x1 and x2 in X1 and X2 , respectively, and j : U1 ∨U2 → X is the inclusion map, then W
Hn (X, x∗ ) = j∗ Hn (U1 ∨ U2 , x∗ )
πn (X, x∗ ).
(1)
N
(ii) If x ∈ X1 \ {x1 } and {x1 } is closed in X1 , then W
Hn (X, x) = Hn (X1 , x)
πn (X, x).
(2)
N
Proof. (i) Equality (1) holds by Theorem 2.4, because every open neighbourhood in the wedge sum is a wedge sum of open neighbourhoods. (ii) Since X1 is a retract of X, one can consider Hn (X1 , x) as a subgroup of Hn (X, x). If {x1 } is closed in X1 , then X1 \ {x1 } is open. Let i : X1 \ {x1 } → X be the inclusion map. By Theorem 2.4, Hn (X, x) = W j∗ Hn (U, x) N πn (X, x), for each neighbourhood U of x. Put U = X1 \ {x1 }. Note that i∗ Hn (X1 \ W {x1 }, x) is a subgroup of Hn (X1 , x). Thus, Hn (X, x) is generated by Hn (X1 , x) ∪ N πn (X, x). Again, W equality (2) holds by normality of N πn (X, x). 2
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Note that the groups in equalities (1) and (2) are not isomorphic, in general. For instance, in Example 3.6 one group is trivial, but not necessarily the other one. A result similar to the following theorem was proved in [1, Theorem 2.5] by a slightly different argument. Corollary 2.6. Let (X1 , x1 ) and (X2 , x2 ) be two pointed spaces, (X, x∗ ) = (X1 , x1 ) ∨ (X2 , x2 ), and n ≥ 1. If X1 and X2 are semilocally strongly contractible at x1 and x2 , respectively, then Hn (X, x∗ ) =
W
πn (X, x∗ ).
N
Proof. Since X1 and X2 are semilocally strongly contractible at x1 and x2 , respectively, there exist open neighbourhoods U1 of x1 and U2 of x2 such that inclusion maps (U1 , x1 ) → (X1 , x1 ) and (U2 , x2 ) → (X2 , x2 ) are nullhomotopic. By joining these homotopies, one can see that j : U1 ∨ U2 → X1 ∨ X2 is nullhomotopic and hence j∗ Hn (U1 ∨ U2 , x∗ ) is trivial. The result holds by Corollary 2.5 (i). 2 The following example shows that Corollary 2.6 does not hold without condition semilocally strongly contractible on both of spaces. Example 2.7. Consider x∗ as the common point of the 1st Griffiths space. We show that if Corollary 2.6 holds for the 1st Griffiths space at x∗ , then π1 (G1 , x∗ ) is trivial which is a contradiction (see [5]). Assume W that H1 (G1 , x∗ ) = N π1 (G1 , x∗ ), then the homomorphism ϕ : Hn (X, x0 ) → N πn (X, x0 ) (see (I)) can be considered as the natural injection. If we show that the homomorphism ϕ is surjective, then the natural W injection N π1 (G1 , x∗ ) → N π1 (G1 , x∗ ) is surjective which is impossible, unless π1 (G1 , x∗ ) is trivial. To prove surjectivity of ϕ, let {[fk ]} ∈ N π1 (G1 , x∗ ) and {Uk }N be a nested local basis at x∗ . Since any n-loop at x∗ is small, we can find a homotopic representative fk of [fk ] in Uk , for all k ∈ N. Now we define f : HE1 → G1 by f |S1k = fk , satisfying ϕ([f ]) = {[fk ]}N . CW-complexes are semilocally strongly contractible at any point. The following result presents an isomorphism for the Hawaiian groups of the wedge sum of CW-complexes. Corollary 2.8. Let n ≥ 2, and X1 , and X2 be two locally finite (n − 1)-connected CW-complexes. If (X, x∗ ) = (X1 , x1 ) ∨ (X2 , x2 ), then Hn (X, x∗ ) ∼ = Hn (X1 , x1 ) ⊕ Hn (X2 , x2 ).
(3)
Proof. CW-complexes are semilocally strongly contractible. Thus by Corollary 2.6, Hn (X, x) ∼ = N πn (X, x), when n ≥ 2. Now by [9, Proposition 6.36], πn (X, x) ∼ = πn (X1 ) ⊕ πn (X2 ), and after a rearrangement, ∼ Hn (X, x) = N πn (X1 ) ⊕ N πn (X2 ). We obtain the result, using [1, Theorem 2.5]. 2 An analogous isomorphism for (3) does not hold, when n = 1. To obtain such an isomorphism on the 1st Hawaiian group, we must replace direct sum by free product, because the fundamental group and the 1st Hawaiian group are not abelian groups, in general. Using Corollary 2.6 and the van Kampen Theorem for wedge sum we have the following result. Corollary 2.9. Let X1 and X2 be two semilocally strongly contractible spaces at x1 and x2 , respectively, and (X, x∗ ) = (X1 , x1 ) ∨ (X2 , x2 ). Then H1 (X, x∗ ) ∼ =
W π1 (X1 , x1 ) ∗ π1 (X2 , x2 ) . N
(4)
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Note that the isomorphism (4) is not similar to the case n ≥ 2, even if X is a CW-complex, unless it is ∼ H1 (X1 , x1 ) ∗ H1 (X2 , x2 ), then by Corollary 2.6, H1 (X, x∗ ) = ∼ simply connected. Suppose that H1 (X, x∗ ) = W W N π1 (X1 , x1 ) ∗ N π1 (X2 , x2 ). Hence, by isomorphism (4) we must have W N
W W π1 (X1 , x1 ) ∗ π1 (X2 , x2 ) ∼ π1 (X1 , x1 ) ∗ π1 (X2 , x2 ), = N
N
which is impossible by [8, Page 183, 6.3.10] if π1 (X1 , x1 ) and π1 (X2 , x2 ) are not trivial. Therefore, we can conclude that H1 (X, x∗ ) H1 (X1 , x1 ) ∗ H1 (X2 , x2 ) if the factors are not trivial. 3. Hawaiian groups of semilocally n-simply connected spaces In this section, we study more on Hawaiian groups of the wedge sum in semilocally n-simply connected spaces. We present results for n = 1 and n ≥ 2, separately, due to the difference in group structures. Recall that for n ≥ 1, a space X is called n-simply connected at x if πn (X, x) is trivial and it is called n-connected at x if πj (X, x) is trivial, for 1 ≤ j ≤ n. Also, X is called semilocally n-simply connected at x if there exists a neighbourhood U of x such that the homomorphism πn (j) : πn (U, x) → πn (X, x), induced by the inclusion, is trivial. Theorem 3.1. Let (X1 , x1 ) and (X2 , x2 ) be two pointed spaces and let (X, x∗ ) = (X1 , x1 ) ∨ (X2 , x2 ). If X is semilocally 1-simply connected at x∗ , then H1 (X, x∗ ) ∼ = j∗ H1 (U1 ∨ U2 ) ×
W
π1 (X, x∗ ),
N
for some neighbourhood U1 ∨ U2 of x∗ with the inclusion map j : U1 ∨ U2 → X. W Proof. By Corollary 2.5 part 1, H1 (X, x∗ ) = j∗ H1 (U1 ∨ U2 , x∗ ) N π1 (X, x∗ ) for each open neighbourhood U1 ∨ U2 of x∗ . Let U1 ∨ U2 be the neighbourhood for which π1 (j) : π1 (U1 ∨ U2 , x∗ ) → π1 (X, x∗ ) is the W trivial homomorphism and let [f ] ∈ j∗ H1 (U1 ∨ U2 , x∗ ) ∩ N π1 (X, x∗ ). Since [f ] ∈ j∗ H1 (U1 ∨ U2 , x∗ ), there exists f˜ : (HE1 , θ) → (U1 ∨ U2 , x∗ ) such that j∗ [f˜] = [f ], or equivalently, j ◦ f˜ f . Also, since W [f ] ∈ N π1 (X, x∗ ), f can be assumed as a map with f |k≥K S1k = C|k≥K S1k for some K ∈ N. Hence j ◦ f˜|k≥K S1k C|k≥K S1k . Using [1, Lemma 2.2], one can replace f˜ with a map fˆ such that fˆ|k≥K S1k W W C|k≥K S1k . Thus [fˆ] ∈ N π1 (U1 ∨ U2 , x∗ ) and then [f ] ∈ N π1 (j)π1 (U1 ∨ U2 , x∗ ) which is trivial, because W W of the choice of U1 ∨U2 . Hence j∗ H1 (U1 ∨U2 , x∗ ) ∩ N π1 (X, x∗ ) = e. Moreover, N π1 (X, x∗ ) is a normal subgroup of H1 (X, x∗ ). Finally, we show that j∗ H1 (U1 ∨ U2 , x∗ ) is normal in H1 (X, x0 ), and therefore, the isomorphism holds. Since X is semilocally 1-simply connected at x∗ , there exist some open neighbourhood U = U1 ∨ U2 of x∗ , such that π1 (j)π1 (U, x∗ ) is trivial in π1 (X, x∗ ). Thus, every 1-loop at x∗ in U is nullhomotopic in X. Let [f ] ∈ H1 (X, x∗ ) and [g] ∈ j∗ H1 (U, x∗ ). We must show that [f ][g][f ]−1 = [f ∗ g ∗ f −1 ] ∈ j∗ H1 (U, x∗ ). Since U is open, there exists K ∈ N such that if k ≥ K, then im(f ∗ g ∗ f −1 )|S1k ⊆ U . Let k < K. Since [g] ∈ j∗ H1 (U, x∗ ), we can consider im(g|S1k ) ⊆ U . Moreover, π1 (j)π1 (U, x∗ ) is trivial in π1 (X, x∗ ). Thus, g|S1k is nullhomotopic in X. Since (f ∗ g ∗ f −1 )|S1k = f |S1k ∗ g|S1k ∗ f −1 |S1k , and g|S1k is nullhomotopic, (f ∗ g ∗ f −1 )|S1k (f ∗ f −1 )|S1k which is nullhomotopic in X. Therefore, one can replace (f ∗ g ∗ f −1 )|S1k by cx0 , for k < K. Moreover, im(f ∗ g ∗ f −1 )|S1k ⊆ U , for k ≥ K, and thus, f ∗ g ∗ f −1 is homotopic to an element of j∗ H1 (U, x∗ ). Hence, [f ] ∗ [g] ∗ [f ]−1 ∈ j∗ H1 (U, x∗ ), that is j∗ H1 (U, x∗ ) is normal in H1 (X, x∗ ) 2 By a similar argument, we can conclude the following result for the wedge sum of 1-simply connected spaces.
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Theorem 3.2. Let (X1 , x1 ) and (X2 , x2 ) be two pointed spaces and let (X, x∗ ) = (X1 , x1 ) ∨ (X2 , x2 ). If π1 (X1 , x) = e for some x ∈ X1 \ {x1 }, then H1 (X, x) ∼ = H1 (X1 , x) ×
W
π1 (X, x).
N
In the following two theorems, we reconstruct isomorphisms in Theorems 3.1 and 3.2 for n ≥ 2. In this case, since all groups are abelian, the direct product notation × must be replaced by the direct sum notation ⊕. Theorem 3.3. Let (X1 , x1 ) and (X2 , x2 ) be two pointed spaces, (X, x∗ ) = (X1 , x1 ) ∨ (X2 , x2 ), and n ≥ 2. If X is semilocally n-simply connected at x∗ , then Hn (X, x∗ ) ∼ = j∗ Hn (U1 ∨ U2 ) ⊕
πn (X, x∗ ),
N
for some neighbourhood U1 ∨ U2 of x∗ with the inclusion map j : U1 ∨ U2 → X. Theorem 3.4. Let (X1 , x1 ) and (X2 , x2 ) be two pointed spaces, (X, x∗ ) = (X1 , x1 ) ∨ (X2 , x2 ), and n ≥ 2. If πn (X1 , x) = e for some x1 = x ∈ X1 , then Hn (X, x) ∼ = Hn (X1 , x) ⊕
πn (X, x).
N
Ž. Virk [10] defined small 1-loop and studied small loop spaces. Note that a nullhomotopic loop is a small loop. H. Passandideh and F.H. Ghane [7] defined and studied the notions of n-homotopically Hausdorffness and small n-loops, for n ≥ 2. An n-loop α : (Sn , 1) → (X, x) is called small if it has a homotopic representative in every open neighbourhood of x. Theorem 3.5. Let (X1 , x1 ) and (X2 , x2 ) be two pointed spaces, (X, x∗ ) = (X1 , x1 ) ∨ (X2 , x2 ), and n ≥ 1. Also, let {x1 } be a closed subset in X1 , and x1 = x ∈ X1 . (i) All n-loops in X at x∗ are small if and only if Hn (X, x∗ ) = j∗ Hn (U1 ∨ U2 , x∗ ),
(5)
for any neighbourhoods U1 and U2 of x1 and x2 in X1 and X2 , respectively, when j : U1 ∨ U2 → X is the inclusion map. (ii) If all n-loops in X at x are small, then Hn (X, x) = Hn (X1 , x).
(6)
Proof. (i) Assume that all n-loops in X at x∗ are small. Let U1 and U2 be two arbitrary neighbourhoods W of x1 and x2 , respectively, and let [f ] ∈ N πn (X, x∗ ). We show that [f ] ∈ j∗ Hn (U1 ∨ U2 , x∗ ), and W then the equality (5) is obtained by Corollary 2.5. Since [f ] ∈ N πn (X, x∗ ), one can consider f as f |k≥K Snk = C|k≥K Snk for some K ∈ N. Moreover, since each n-loop in X is small at x∗ , any n-loop α is homotopic to some n-loop in U1 ∨ U2 , say α ˜ : Sn → U1 ∨ U2 , such that j ◦ α ˜ α. By induction on finite join of n-loops, f | k
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Fig. 2. The wedge sum of a circle and a cone on the Hawaiian earring.
Conversely, let α be an n-loop in X at x∗ . Consider the map f : (HEn , θ) → (X, x∗ ) so that f |Sn1 = α and f |Snk = c for k > 1. Then [f ] ∈ j∗ Hn (U1 ∨ U2 , x∗ ) by equality (5) for any neighbourhoods U1 and U2 . Let [f˜] be the element of Hn (U1 ∨ U2 , x∗ ) such that j ◦ f˜ = f . Hence, j ◦ f˜|Sn1 = f |Sn1 = α, that is α is homotopic to some n-loop in U1 ∨ U2 . Since U1 and U2 are arbitrary neighbourhoods, α is a small n-loop. (ii) Since X1 \ {x1 } is open and all n-loops at x in X are small, similar to the proof of the previous part, W one can show that N πn (X, x) ⊆ i∗ Hn (X1 \ {x1 }, x), where i : X1 \ {x1 } → X is the inclusion map. Moreover, i∗ Hn (X1 \ {x1 }, x) is contained in Hn (X1 , x) as a subgroup. Thus, by Corollary 2.5, the equality (6) holds. 2 Since a nullhomotopic n-loop is a special case of small n-loop, the equalities (5) and (6) hold for n-simply connected spaces. For example, Theorem 3.5 holds for two cones joining at their vertices which is not only n-simply connected, but also contractible. Recall that the 1st Griffiths space, the wedge sum of two cones, is not contractible, even more Griffiths [5] proved that it is not 1-simply connected. Example 3.6. For a space X, put Y = X×[−1,1] X×{0} the wedge sum of two cones over X at their vertices and x ˜t = [(x0 , t)] for t = 0. One can see that Y is contractible at the common point, and hence, it is n-simply H (X,x0 ) connected. By Theorem 3.5, Hn (Y, x ˜t ) = Hn (CX, x ˜t ). Also, by [1, Theorem 2.13], Hn (Y, x0 ) ∼ . = W n π (X,x ) k∈N
n
0
Let x∗ be the common vertex of the two cones. Then Hn (Y, x∗ ) is trivial, for Y is semilocally strongly W contractible at x∗ . By using Corollary 2.6, Hn (Y, x∗ ) = N πn (Y, x∗ ) which is trivial. The following example reveals that Theorem 3.5 does not hold, if some non small loop exists. Example 3.7. Let (X, x∗ ) = (C(HE1 ), θ) ∨ (S1 , 1) (see Fig. 2). If one assumes that H1 (X, x∗ ) = H1 (C(HE1 ), x∗ ), then the simple 1-loop in S1 must be nullhomotopic which is a contradiction. 4. Hawaiian groups of Griffiths spaces In this section, by generalizing the Griffiths space to higher dimensions and applying the results of Sections 2 and 3, we study the nth Hawaiian group of the nth Griffiths space, for n ≥ 1. Eda [3] introduced the free σ-product ×σN Z as the group consisting of all reduced σ-words, and then proved that it is isomorphic to π1 (HE1 , θ) [3, Theorem A.1]. To prove, Eda remarked that each 1-loop in the 1-dimensional Hawaiian earring is homotopic to some proper 1-loop [3, Lemma A.3]. A 1-loop α : ˙ → (HE1 , θ) is called proper whenever for each subinterval [a, b] ⊆ I, if α|[a,b] is nullhomotopic, then it (I, I) is constant. Also, O. Bogopolski and A. Zastrow proved that π1 (G1 , a) ∼ =
×σN Z ×σNe Z, ×σNo Z
N
,
(7)
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where No and Ne denote the set of odd and even numbers, respectively [2, Theorem 3.4]. This isomorphism is induced by the natural embedding ι : HE1 → G1 causing epimorphism π1 (ι) : ×σN Z → π1 (G1 , x∗ ) together N
with ×σNe Z, ×σNo Z as its kernel. The following lemma gives a useful description of the group H1 (HE1 , θ) which is used in sequel. Lemma 4.1. Let B be the subgroup of N ×σN Z consisting of all countably infinite tuples of reduced σ-words such that the number of components including letter of type m is finite, for all m ∈ N. Then H1 (HE1 , θ) ∼ = B. Proof. By [1, Theorem 2.9], group H1 (HE1 , θ) is isomorphic to the subgroup of N π1 (HE1 , θ) consisting of all sequences of homotopy classes of 1-loops with some representative converging uniformly to the constant 1-loop. We show that B equals this subgroup. Let {Ul ; l ∈ N} be the local basis at θ defined in the proof of [1, Theorem 2.9], and also let {[fk ]} ∈ 1 N π1 (HE , θ). Then {fk } converges uniformly to the constant 1-loop if and only if for each l ∈ N, there exists Kl ∈ N such that im(fk ) ⊆ Ul whenever k ≥ Kl . Assume that fk is the corresponding proper representative for all k ∈ N. The image of fk is contained in Ul if and only if im(fk ) ∩ S1m = {θ}, for all m < l, as for such m the intersection U ∩ S1m is contractible and fk has no trivial subpath. Therefore, {fk } converges uniformly to the constant 1-loop if and only if for each l ∈ N, there exists Kl ∈ N such that im(fk ) ∩ S1m = {θ} whenever k ≥ Kl . By [3, Theorem A.1], π1 (HE1 , θ) ∼ = ×σN Z, the group of reduced σ-words. Moreover, in a given reduced σ-word, the letter of type m exists if and only if the mth circle S1m of HE1 appears in the corresponding proper 1-loop. Therefore H1 (HE1 , θ) is isomorphic to the subgroup of N ×σN Z consisting of all countably infinite tuples of reduced σ-words such that the number of components including letter of type m is finite, for all m ∈ N. 2 The following theorem investigates the structure of the 1st Hawaiian group of the 1st Griffiths space at the common point, the two vertices and the other points. Let i : HE1 → G1 \ {v1 , v2 } be the embedding which maps (2m − 1)th circle onto the horizontal left mth circle and maps 2mth circle onto the horizontal right mth circle, for m ∈ N (see Fig. 1). Also, let j : G1 \ {v1 , v2 } → G1 be the inclusion map. Theorem 4.2. Let G1 be the 1st Griffiths space, x∗ the common point, a ∈ A ∪A , and x ∈ G1 \(A ∪A ∪{x∗ }). Then H1 (G1 , x∗ ) ∼ =
H1 (HE1 , θ) ∼ B , = N W σ σ Z i−1 ker j ∗ ∗ × Z, × Ne No N
W W H1 (HE1 , θ) B ×σN Z ∼ H1 (G1 , a) ∼ π (G , a) , × × = W = 1 1 W N 1 σ ×σNe Z, ×σNo Z N π1 (HE , θ) N ×N Z N N H1 (G1 , x) ∼ =
W N
×σN Z ×σNe Z, ×σNo Z
N
.
(i)
(ii)
(iii)
Proof. (i) Since all 1-loops at x∗ are small, Theorem 3.5 implies that H1 (G1 , x∗ ) = j∗ H1 (U1 ∨ U2 , x∗ ), when U1 and U2 are arbitrary neighbourhoods of x∗ in the two cones. Suppose that Um (m = 1, 2) is the whole of the corresponding cone except its vertex. Then U1 ∨ U2 = G1 \ {v1 , v2 } and thus H1 (G1 , x∗ ) = j∗ H1 (U1 ∨ U2 , x∗ ) = j∗ H1 (G1 \ {v1 , v2 }, x∗ ).
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Moreover, the embedded Hawaiian earring i(HE1 ) is a deformation retract of G1 \{v1 , v2 } with projection p : G1 \ {v1 , v2 } → HE1 as the retraction. Therefore, ∼ j∗ i∗ H1 (HE1 , θ). H1 (G1 , x∗ ) = j∗ H1 (G1 \ {v1 , v2 }, x∗ ) = By the First Isomorphism Theorem, i∗ H1 (HE1 , θ) ∼ H1 (HE1 , θ) j∗ i∗ H1 (HE1 , θ) ∼ . = = −1 ker j∗ i∗ ker j∗ W σ N σ In the following, we show that i−1 ∗ ker j∗ is mapped isomorphically onto group N ×Ne Z, ×No Z , by the same isomorphism which maps H1 (HE1 , θ) onto B, in Lemma 4.1. Let [g] ∈ i−1 ∗ ker j∗ . Then i∗ [g] ∈ ker j∗ , and therefore, j ◦i ◦g Cx∗ . Hence, the 1-loops (j ◦i ◦g|S1k )’s are nullhomotopic with some null convergent sequence of homotopies, say {Hk }. Thus, there exists K ∈ N such that imHk ⊆ G1 \ {v1 , v2 } for k ≥ K. Therefore, i ◦ g|S1k is nullhomotopic in G1 \ {v1 , v2 } for k ≥ K by null convergent homotopies {Hk }k≥K . Accordingly, i ◦ g|k≥K S1k is nullhomotpic in G \ {v1 , v2 }. Therefore, p ◦ i ◦ g|k≥K S1k = g|k≥K S1k is nullhomotopic in HE1 . Moreover, for k < K, j ◦ i ◦ g|Sk1 is nullhomotopic in G1 or equivalently [g|Sk1 ] ∈ ker π1 (ι). Note that N
ker π1 (ι) is mapped isomorphically onto ×σNe Z, ×σNo Z , by the same isomorphism mapping π1 (HE, θ) W N onto ×σN Z. Thus, [g] is corresponded injectively to an element of N ×σNe Z, ×σNo Z , where the correspondence is the same as the isomorphism mapping H1 (HE, θ) onto B, in Lemma 4.1. One can check that this correspondence is also surjective. (ii) By Theorem 3.2, H1 (G1 , a) ∼ = H1 (CHE1 , a) ×
W
π1 (G1 , a),
N
and by isomorphism (7), H1 (G1 , a) ∼ = H1 (CHE1 , a) ×
W
×σN Z N
N
×σNe Z, ×σNo Z
.
H (HE ,θ) Now by [1, Theorem 2.13] H1 (CHE1 , a) ∼ = W 1π1 (HE1 ,θ) . N W W By Lemma 4.1, H1 (HE1 , θ) ∼ = B, which maps subgroup N π1 (HE1 , θ) onto N ×σN Z. Consequently H1 (CHE1 , a) ∼ = WB×σ Z , and hence the isomorphism (ii) holds. N N W (iii) Obviously, G1 is semilocally strongly contractible at x. Therefore, H1 (G1 , x) ∼ = N π1 (G1 , x). 2 1
Eda and Kawamura [4] defined the nth Griffiths space by the wedge sum of two copies of cones on HEn at the origin for n ≥ 2. Furthermore, they proved that πn (Gn ) is trivial, for n ≥ 2 [4, Corollary 1.4]. In the following theorem we give some information on the structure of the nth Hawaiian group of the nth Griffiths space. Note that by Zk we mean the subgroup of N Z consisting of all countably infinite tuples with all zero components except possibly for any collection of the first k components, and thus Z1 Z2 Z3 · · · . Theorem 4.3. Let n ≥ 2, Gn be the nth Griffiths space, and a ∈ A ∪ A ∪ {x∗ }. Then Z Hn (HEn , θ) ∼ ∼ Hn (Gn , a) = = N N k , n N πn (HE , θ) k∈N NZ
(i)
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(ii) and Hn (Gn , x) is trivial, where x ∈ G1 \ (A ∪ A ∪ {x∗ }). Proof. (i) If a ∈ A ∪ A , then by Theorem 3.4, Hn (Gn , a) ∼ = Hn (CHEn , a) ⊕ N πn (Gn , a). Since πn (Gn , a) is trivial [4, Corollary 1.4], the second factor is deleted, and hence Hn (Gn , a) ∼ = H (CHEn , a). Using n n H (HE ,θ) n n n ∼ [1, Theorem 2.13], Hn (CHE , a) = πn (HEn ,θ) . Replacing Hn (HE , θ) with N N Z via [1, Theorem N 2.11], which maps N πn (HEn , θ) isomorphically onto the subgroup k∈N N Zk , the result holds. Let a = x∗ . By Corollary 2.5, Hn (Gn , x∗ ) = j∗ Hn (U1 ∨ U2 , x∗ ) N πn (Gn , x∗ ) for any neighbourhoods U1 and U2 of the origin in two cones, when j : U1 ∨ U2 → Gn is the inclusion. Since πn (Gn ) is trivial, Hn (Gn , x∗ ) = j∗ Hn (U1 ∨ U2 , x∗ ) for any neighbourhoods U1 and U2 . Put U1 to be the whole of one cone without its vertex, and U2 be the another cone without its vertex. Then U1 ∨ U2 = Gn \ {v, v }. Similar to the proof of Theorem 4.2, Part 1, one can replace j∗ Hn (U1 ∨ U2 , x∗ ) by j∗ i∗ Hn (HEn , θ) in the above isomorphism, where i : HEn → Gn \ {v, v } is the natural embedding. Hence Hn (Gn , x∗ ) ∼ = Hn (HEn ,θ) −1 . Similar to the proof of Theorem 4.2, Part 1, one can prove that i ker j = ker π (ι). ∗ n ∗ N i−1 ∗ ker j∗ Hn (HEn ,θ) n ∼ Since πn (Gn ) is trivial, ker πn (ι) = πn (HE , θ). Therefore, Hn (Gn , x∗ ) = . In the case n
N
πn (HE ,θ)
,θ) ∼ N N Z a ∈ A ∪ A , we show that Hnπ(HE = n k as required. N n (HE ,θ) k∈N N Z (ii) Since Gn is semilocally strongly contractible at x, Hn (Gn , x) ∼ = N πn (Gn ) by Corollary 2.6. Moreover, πn (Gn ) is trivial by [4, Corollary], and hence Hn (Gn , x) is trivial. 2 n
Acknowledgements The authors would like to thank the referee for his/her careful reading and useful comments and suggestions that helped to improve the article. This research was supported by a grant from Ferdowsi University of Mashhad-Graduate Studies (No. 42705). References [1] A. Babaee, B. Mashayekhy, H. Mirebrahimi, On Hawaiian groups of some topological spaces, Topol. Appl. 159 (8) (2012) 2043–2051. [2] O. Bogopolski, A. Zastrow, The word problem for some uncountable groups given by countable words, Topol. Appl. 159 (2012) 569–586. [3] K. Eda, Free σ products and noncommutatively slender groups, J. Algebra 148 (1992) 243–263. [4] K. Eda, K. Kawamura, Homotopy and homology groups of the n-dimensional Hawaiian earring, Fundam. Math. 165 (1) (2000) 17–28. [5] H.B. Griffiths, The fundamental group of two spaces with a common point, Q. J. Math., Oxford Ser. (2) 5 (1954) 175–190; H.B. Griffiths, correction: Q. J. Math., Oxford Ser. (2) 6 (1955) 154–155. [6] U.H. Karimov, D. Repovš, Hawaiian groups of topological spaces (Russian), Usp. Mat. Nauk 61 (5) (2006) 185–186; transl. in Russian: Russ. Math. Surv. 61 (5) (2006) 987–989. [7] H. Passandideh, F.H. Ghane, Homotopy properties of subsets of Euclidean spaces, Topol. Appl. 194 (2015) 202–211. [8] D.J.S. Robinson, A Course in the Group Theory, Springer-Verlag, New York, 1996. [9] R.M. Switzer, Algebraic Topology–Homotopy and Homology, Springer-Verlag, Berlin, 1975. [10] Ž. Virk, Small loop spaces, Topol. Appl. 157 (2010) 451–455.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
On a Van Kampen theorem for Hawaiian groups Ameneh Babaee, Behrooz Mashayekhy ∗ , Hanieh Mirebrahimi, Hamid Torabi Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhad, Iran
a r t i c l e
i n f o
Article history: Received 13 July 2017 Received in revised form 17 March 2018 Accepted 8 April 2018 Available online 10 April 2018 MSC: 55Q05 55Q20 55P65 55Q52
a b s t r a c t The paper is devoted to study the nth Hawaiian group Hn , n ≥ 1, of the wedge sum of two spaces (X, x∗ ) = (X1 , x1 ) ∨(X2 , x2 ). We are going to give some versions of the van Kampen theorem for Hawaiian groups of the wedge sum of spaces. First, among some results on Hawaiian groups of semilocally strongly contractible spaces, we present a structure for the nth Hawaiian group of the wedge sum of CW-complexes. Second, we give more informative structures for the nth Hawaiian group of the wedge sum X, when X is semilocally n-simply connected at x∗ . Finally, as a consequence, we study Hawaiian groups of Griffiths spaces for all dimensions n ≥ 1 to give some information about their structure at any points. © 2018 Elsevier B.V. All rights reserved.
Keywords: Hawaiian group Hawaiian earring Van Kampen theorem Griffiths space
1. Introduction and motivation In 2006, U.H. Karimov and D. Repovš [6] defined the nth Hawaiian group as a covariant functor Hn : hT op∗ → Groups from the pointed homotopy category, hT op∗ , to the category of all groups, Groups, for n ≥ 1. For any pointed space (X, x0 ), the nth Hawaiian group Hn (X, x0 ) was defined to be the set of all pointed homotopy classes [f ], where f : (HEn , θ) → (X, x0 ) is a pointed map. Here, HEn denotes the n-dimensional Hawaiian earring, the union of n-spheres Snk in Rn+1 with centre (1/k, 0, . . . , 0) and radius 1/k, and the origin θ is considered as the base point. The operation of the nth Hawaiian group arises component-wisely from the operation of the nth homotopy group. Thus the following map ϕ : Hn (X, x0 ) → πn (X, x0 ), (I) N
* Corresponding author. E-mail addresses: [email protected] (A. Babaee), [email protected] (B. Mashayekhy), [email protected] (H. Mirebrahimi), [email protected] (H. Torabi). https://doi.org/10.1016/j.topol.2018.04.005 0166-8641/© 2018 Elsevier B.V. All rights reserved.
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defined by ϕ([f ]) = ([f |Sn1 ], [f |Sn2 ], ...) is a homomorphism, for all n ∈ N. Note that in general, the homomorphism ϕ is not injective nor surjective. Let C(HE1 ) be the cone over 1-dimensional Hawaiian earring. Since C(HE1 ) is simply connected and H1 (C(HE1 ), θ) is not trivial [6], ϕ can not be injective. Moreover, W for every locally n-simply connected first countable space X, im(ϕ) = N πn (X, x0 ) [6, Theorem 1] which is a proper subgroup of N πn (X, x0 ) whenever πn (X, x0 ) is nontrivial. Here, we consider the weak direct W product N πn (X, x0 ) as a subgroup of Hn (X, x0 ), because it is proved that for every pointed space (X, x0 ), W N πn (X, x0 ) can be embedded as a normal subgroup in Hn (X, x0 ) (see [1, Lemma 2.4] and [1, Theorem 2.13]). Although the nth Hawaiian group functor is a pointed homotopy invariant functor on the category of all pointed topological spaces, it is not freely homotopy invariant. Because unlike other homotopy invariant functors, Hawaiian groups of contractible spaces are not necessarily trivial. Karimov and Repovš [6] gave a contractible space, the cone over HE1 , with nontrivial 1st Hawaiian group at some points (consisting of the points at which CHE1 is not locally 1-simply connected), but with trivial homotopy, homology and cohomology groups. More precisely, it can be shown that H1 (C(HE1 ), θ) is uncountable, using [6, Theorem 2]: “if X has a countable local basis at x0 , then countability of nth Hawaiian group Hn (X, x0 ) implies locally n-simply connectedness of X at x0 .” Furthermore, a converse of the above statement can be found in [1, Corollaries 2.16 and 2.17]: “let X have a countable local basis at x0 . Then Hn (CX, x ˜) is trivial if and only if X is locally n-simply connected at x0 and it is uncountable otherwise”. In addition, unlike homotopy groups, Hawaiian groups of pointed space (X, x0 ) depend on the behaviour of X at x0 , and then their structures depend on the choice of base point. In this regard, there exist some examples of path connected spaces with non-isomorphic Hawaiian groups at different points, such as the n-dimensional Hawaiian earring, where n ≥ 2 (see [1, Corollary 2.11]). Despite the above different behaviours between Hawaiian groups and homotopy groups, they have some similar behaviours. For instance, it was proved that similar to the nth homotopy group, the nth Hawaiian group of any pointed space is abelian, for all n ≥ 2 [1, Theorem 2.3]. Also, the Hawaiian groups preserve products in the category hT op∗ [1, theorem 2.12]. In this paper, we investigate the Hawaiian groups of the coproduct in the category hT op∗ which is the wedge sum of a given family of pointed spaces. In fact, we are going to give some versions of the van Kampen theorem for Hawaiian groups of the wedge sum of spaces. In Section 2, among some results on Hawaiian groups of semilocally strongly contractible spaces, we intend to present a structure for the nth Hawaiian group of the wedge sum of CW-complexes. A given space X is semilocally strongly contractible at x0 if the inclusion i : U → X is nullhomotopic in X relative to {x0 }, for some open neighbourhood U of x0 (see [4]). In Section 3, we present the nth Hawaiian group of the wedge sum (X, x∗) = (X1 , x1 ) ∨ (X2 , x2 ) as the direct product of two its subgroups that are more perceptible, when X is semilocally n-simply connected at x∗ . Also, we prove that the Hawaiian group of a pointed space equals the Hawaiian group of every neighbourhood of the base point if all n-loops are small. An n-loop α : (Sn , 1) → (X, x) is called small, if for each neighbourhood U of x, α has a homotopic representative in U (see [7,10]). In Section 4, we investigate the Hawaiian groups of the nth Griffiths space, introduced in [4, Corollary 1.4], by generalizing the well-known Griffiths space, as the wedge sum of two copies of the cone over the n-dimensional Hawaiian earring. For the sake of clarity, we call the well known Griffiths space as the 1st Griffiths space. Then, using results of Sections 2 and 3, we intend to give some information about the structure of Hawaiian groups of Griffiths spaces at any points. In this paper, all homotopies are relative to the base point, unless stated otherwise.
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Fig. 1. The 1st Griffiths space.
2. Hawaiian groups of semilocally strongly contractible spaces In this section, we investigate Hawaiian groups of wedge sum of pointed spaces which are semilocally strongly contractible at the base points. The property semilocally strongly contractibility was defined by Eda and Kawamura [4]. Definition 2.1. A space X is called semilocally strongly contractible at x0 if there exists some open neighbourhood U of x0 such that the inclusion map i : U → X is nullhomotopic in X relative to {x0 }, in other words, i : (U, x0 ) → (X, x0 ) is nullhomotopic. First, we compare semilocally strongly contractible property with some familiar properties, such as locally contractible, locally strongly contractible and semilocally contractible properties. Recall 2.2. Let (X, x) be a pointed space, then X is called locally contractible at x if for each open neighbourhood U of x in X, there exists some open neighbourhood V of x contained in U such that the inclusion V → U is freely nullhomotopic. We say that X is locally strongly contractible at x if for each open neighbourhood U of x in X, there exists some open neighbourhood V of x contained in U such that the inclusion V → U is nullhomotopic relative to {x}, or briefly, (V, x) → (U, x) is nullhomotopic. Moreover, X is called semilocally contractible at x, if there exists some open neighbourhood U of x such that the inclusion i : U → X is freely nullhomotopic in X. One can see that there are some relations between these properties. Locally strongly contractible property implies locally contractibility, and locally contractible property implies semilocally contractibility, but the converse statements do not necessarily hold. Moreover, locally strongly contractible property implies semilocally strongly contractibility, and semilocally strongly contractible implies semilocally contractibility, but converse statements do not necessarily hold. The following example shows that a given space can behave differently at different points. Example 2.3. The 1st Griffiths space G1 is locally strongly contractible at two vertices v and v . Therefore, it is locally contractible, semilocally strongly contractible and semilocally contractible at two vertices v and v . Moreover, G1 is semilocally contractible at all points except at the common point x∗ . Also, G1 is semilocally contractible at any point of A and A , but not semilocally strongly contractible, where A and A are open
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intervals from the common point x∗ to the corresponding vertices. Finally, G1 is neither semilocally strongly contractible, semilocally contractible nor even semilocally 1-simply connected at x∗ (see [5]). The 1-dimensional Hawaiian earring HE1 is not semilocally contractible at the origin. However, it is locally strongly contractible at other points. There exist many spaces which are semilocally contractible at each point, but not strongly at some points. For example, consider CΔ as the cone over the Seirpeinski gasket. It is semilocally contractible at any points, but it is semilocally strongly contractible just at the vertex. In the following theorem, we prove that Hawaiian group is generated by two subgroups depend on local behaviour at the given point. It helps us to investigate Hawaiian groups of spaces having specific local properties. Theorem 2.4. Let (X, x0 ) be a pointed space and n ≥ 1. Then Hn (X, x0 ) = j∗ Hn (U, x0 ) each open neighbourhood U with x0 ∈ U ⊆ X and j : U → X as the inclusion map.
W N
πn (X, x0 ), for
Proof. Let U be any open neighbourhood of x0 , and j : U → X be the inclusion map. Also, let f : (HEn , θ) → (X, x0 ) be a pointed map. Since U is open in X, there exists K ∈ N such that if k ≥ K, then im(f |k≥K Snk ) ⊆ U . We can define f , f : (HEn , θ) → (X, x0 ) by f |k
C|k≥K Snk , f |k
at x0 . Recall that for any classes, [f ] ∗ [g] is obtained by (f ∗ g)|Snk = f |Snk ∗ g|Snk . Hence, [f ] = [f ][f ] = [f ][f ]. W Moreover, [f ] is an element of N πn (X, x∗ ), and also, [f ] is an element of j∗ Hn (U, x0 ). Therefore, Hn (X, x∗ ) W W is generated by j∗ Hn (U, x0 ) ∪ N πn (X, x0 ). Since N πn (X, x0 ) is a normal subgroup of Hn (X, x0 ) (see the proof of [1, Theorem 2.13]), the equality holds. 2 The following lemma describes the Hawaiian group of the wedge sum of two spaces by its subgroups. This construction helps us to obtain next isomorphisms. Let {Gi }i∈I be a family of indexed groups and ei W be the identity element of Gi . Then by i∈I Gi , we mean the weak direct product of the family {Gi }i∈I consisting of all elements {gi }i∈I of i∈I Gi such that gi = ei , for all i ∈ I except a finite number. Corollary 2.5. Let (X1 , x1 ) and (X2 , x2 ) be two pointed spaces, (X, x∗ ) = (X1 , x1 ) ∨ (X2 , x2 ), and n ≥ 1. (i) If U1 and U2 are two open neighbourhoods of x1 and x2 in X1 and X2 , respectively, and j : U1 ∨U2 → X is the inclusion map, then W
Hn (X, x∗ ) = j∗ Hn (U1 ∨ U2 , x∗ )
πn (X, x∗ ).
(1)
N
(ii) If x ∈ X1 \ {x1 } and {x1 } is closed in X1 , then W
Hn (X, x) = Hn (X1 , x)
πn (X, x).
(2)
N
Proof. (i) Equality (1) holds by Theorem 2.4, because every open neighbourhood in the wedge sum is a wedge sum of open neighbourhoods. (ii) Since X1 is a retract of X, one can consider Hn (X1 , x) as a subgroup of Hn (X, x). If {x1 } is closed in X1 , then X1 \ {x1 } is open. Let i : X1 \ {x1 } → X be the inclusion map. By Theorem 2.4, Hn (X, x) = W j∗ Hn (U, x) N πn (X, x), for each neighbourhood U of x. Put U = X1 \ {x1 }. Note that i∗ Hn (X1 \ W {x1 }, x) is a subgroup of Hn (X1 , x). Thus, Hn (X, x) is generated by Hn (X1 , x) ∪ N πn (X, x). Again, W equality (2) holds by normality of N πn (X, x). 2
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Note that the groups in equalities (1) and (2) are not isomorphic, in general. For instance, in Example 3.6 one group is trivial, but not necessarily the other one. A result similar to the following theorem was proved in [1, Theorem 2.5] by a slightly different argument. Corollary 2.6. Let (X1 , x1 ) and (X2 , x2 ) be two pointed spaces, (X, x∗ ) = (X1 , x1 ) ∨ (X2 , x2 ), and n ≥ 1. If X1 and X2 are semilocally strongly contractible at x1 and x2 , respectively, then Hn (X, x∗ ) =
W
πn (X, x∗ ).
N
Proof. Since X1 and X2 are semilocally strongly contractible at x1 and x2 , respectively, there exist open neighbourhoods U1 of x1 and U2 of x2 such that inclusion maps (U1 , x1 ) → (X1 , x1 ) and (U2 , x2 ) → (X2 , x2 ) are nullhomotopic. By joining these homotopies, one can see that j : U1 ∨ U2 → X1 ∨ X2 is nullhomotopic and hence j∗ Hn (U1 ∨ U2 , x∗ ) is trivial. The result holds by Corollary 2.5 (i). 2 The following example shows that Corollary 2.6 does not hold without condition semilocally strongly contractible on both of spaces. Example 2.7. Consider x∗ as the common point of the 1st Griffiths space. We show that if Corollary 2.6 holds for the 1st Griffiths space at x∗ , then π1 (G1 , x∗ ) is trivial which is a contradiction (see [5]). Assume W that H1 (G1 , x∗ ) = N π1 (G1 , x∗ ), then the homomorphism ϕ : Hn (X, x0 ) → N πn (X, x0 ) (see (I)) can be considered as the natural injection. If we show that the homomorphism ϕ is surjective, then the natural W injection N π1 (G1 , x∗ ) → N π1 (G1 , x∗ ) is surjective which is impossible, unless π1 (G1 , x∗ ) is trivial. To prove surjectivity of ϕ, let {[fk ]} ∈ N π1 (G1 , x∗ ) and {Uk }N be a nested local basis at x∗ . Since any n-loop at x∗ is small, we can find a homotopic representative fk of [fk ] in Uk , for all k ∈ N. Now we define f : HE1 → G1 by f |S1k = fk , satisfying ϕ([f ]) = {[fk ]}N . CW-complexes are semilocally strongly contractible at any point. The following result presents an isomorphism for the Hawaiian groups of the wedge sum of CW-complexes. Corollary 2.8. Let n ≥ 2, and X1 , and X2 be two locally finite (n − 1)-connected CW-complexes. If (X, x∗ ) = (X1 , x1 ) ∨ (X2 , x2 ), then Hn (X, x∗ ) ∼ = Hn (X1 , x1 ) ⊕ Hn (X2 , x2 ).
(3)
Proof. CW-complexes are semilocally strongly contractible. Thus by Corollary 2.6, Hn (X, x) ∼ = N πn (X, x), when n ≥ 2. Now by [9, Proposition 6.36], πn (X, x) ∼ = πn (X1 ) ⊕ πn (X2 ), and after a rearrangement, ∼ Hn (X, x) = N πn (X1 ) ⊕ N πn (X2 ). We obtain the result, using [1, Theorem 2.5]. 2 An analogous isomorphism for (3) does not hold, when n = 1. To obtain such an isomorphism on the 1st Hawaiian group, we must replace direct sum by free product, because the fundamental group and the 1st Hawaiian group are not abelian groups, in general. Using Corollary 2.6 and the van Kampen Theorem for wedge sum we have the following result. Corollary 2.9. Let X1 and X2 be two semilocally strongly contractible spaces at x1 and x2 , respectively, and (X, x∗ ) = (X1 , x1 ) ∨ (X2 , x2 ). Then H1 (X, x∗ ) ∼ =
W π1 (X1 , x1 ) ∗ π1 (X2 , x2 ) . N
(4)
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Note that the isomorphism (4) is not similar to the case n ≥ 2, even if X is a CW-complex, unless it is ∼ H1 (X1 , x1 ) ∗ H1 (X2 , x2 ), then by Corollary 2.6, H1 (X, x∗ ) = ∼ simply connected. Suppose that H1 (X, x∗ ) = W W N π1 (X1 , x1 ) ∗ N π1 (X2 , x2 ). Hence, by isomorphism (4) we must have W N
W W π1 (X1 , x1 ) ∗ π1 (X2 , x2 ) ∼ π1 (X1 , x1 ) ∗ π1 (X2 , x2 ), = N
N
which is impossible by [8, Page 183, 6.3.10] if π1 (X1 , x1 ) and π1 (X2 , x2 ) are not trivial. Therefore, we can conclude that H1 (X, x∗ ) H1 (X1 , x1 ) ∗ H1 (X2 , x2 ) if the factors are not trivial. 3. Hawaiian groups of semilocally n-simply connected spaces In this section, we study more on Hawaiian groups of the wedge sum in semilocally n-simply connected spaces. We present results for n = 1 and n ≥ 2, separately, due to the difference in group structures. Recall that for n ≥ 1, a space X is called n-simply connected at x if πn (X, x) is trivial and it is called n-connected at x if πj (X, x) is trivial, for 1 ≤ j ≤ n. Also, X is called semilocally n-simply connected at x if there exists a neighbourhood U of x such that the homomorphism πn (j) : πn (U, x) → πn (X, x), induced by the inclusion, is trivial. Theorem 3.1. Let (X1 , x1 ) and (X2 , x2 ) be two pointed spaces and let (X, x∗ ) = (X1 , x1 ) ∨ (X2 , x2 ). If X is semilocally 1-simply connected at x∗ , then H1 (X, x∗ ) ∼ = j∗ H1 (U1 ∨ U2 ) ×
W
π1 (X, x∗ ),
N
for some neighbourhood U1 ∨ U2 of x∗ with the inclusion map j : U1 ∨ U2 → X. W Proof. By Corollary 2.5 part 1, H1 (X, x∗ ) = j∗ H1 (U1 ∨ U2 , x∗ ) N π1 (X, x∗ ) for each open neighbourhood U1 ∨ U2 of x∗ . Let U1 ∨ U2 be the neighbourhood for which π1 (j) : π1 (U1 ∨ U2 , x∗ ) → π1 (X, x∗ ) is the W trivial homomorphism and let [f ] ∈ j∗ H1 (U1 ∨ U2 , x∗ ) ∩ N π1 (X, x∗ ). Since [f ] ∈ j∗ H1 (U1 ∨ U2 , x∗ ), there exists f˜ : (HE1 , θ) → (U1 ∨ U2 , x∗ ) such that j∗ [f˜] = [f ], or equivalently, j ◦ f˜ f . Also, since W [f ] ∈ N π1 (X, x∗ ), f can be assumed as a map with f |k≥K S1k = C|k≥K S1k for some K ∈ N. Hence j ◦ f˜|k≥K S1k C|k≥K S1k . Using [1, Lemma 2.2], one can replace f˜ with a map fˆ such that fˆ|k≥K S1k W W C|k≥K S1k . Thus [fˆ] ∈ N π1 (U1 ∨ U2 , x∗ ) and then [f ] ∈ N π1 (j)π1 (U1 ∨ U2 , x∗ ) which is trivial, because W W of the choice of U1 ∨U2 . Hence j∗ H1 (U1 ∨U2 , x∗ ) ∩ N π1 (X, x∗ ) = e. Moreover, N π1 (X, x∗ ) is a normal subgroup of H1 (X, x∗ ). Finally, we show that j∗ H1 (U1 ∨ U2 , x∗ ) is normal in H1 (X, x0 ), and therefore, the isomorphism holds. Since X is semilocally 1-simply connected at x∗ , there exist some open neighbourhood U = U1 ∨ U2 of x∗ , such that π1 (j)π1 (U, x∗ ) is trivial in π1 (X, x∗ ). Thus, every 1-loop at x∗ in U is nullhomotopic in X. Let [f ] ∈ H1 (X, x∗ ) and [g] ∈ j∗ H1 (U, x∗ ). We must show that [f ][g][f ]−1 = [f ∗ g ∗ f −1 ] ∈ j∗ H1 (U, x∗ ). Since U is open, there exists K ∈ N such that if k ≥ K, then im(f ∗ g ∗ f −1 )|S1k ⊆ U . Let k < K. Since [g] ∈ j∗ H1 (U, x∗ ), we can consider im(g|S1k ) ⊆ U . Moreover, π1 (j)π1 (U, x∗ ) is trivial in π1 (X, x∗ ). Thus, g|S1k is nullhomotopic in X. Since (f ∗ g ∗ f −1 )|S1k = f |S1k ∗ g|S1k ∗ f −1 |S1k , and g|S1k is nullhomotopic, (f ∗ g ∗ f −1 )|S1k (f ∗ f −1 )|S1k which is nullhomotopic in X. Therefore, one can replace (f ∗ g ∗ f −1 )|S1k by cx0 , for k < K. Moreover, im(f ∗ g ∗ f −1 )|S1k ⊆ U , for k ≥ K, and thus, f ∗ g ∗ f −1 is homotopic to an element of j∗ H1 (U, x∗ ). Hence, [f ] ∗ [g] ∗ [f ]−1 ∈ j∗ H1 (U, x∗ ), that is j∗ H1 (U, x∗ ) is normal in H1 (X, x∗ ) 2 By a similar argument, we can conclude the following result for the wedge sum of 1-simply connected spaces.
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Theorem 3.2. Let (X1 , x1 ) and (X2 , x2 ) be two pointed spaces and let (X, x∗ ) = (X1 , x1 ) ∨ (X2 , x2 ). If π1 (X1 , x) = e for some x ∈ X1 \ {x1 }, then H1 (X, x) ∼ = H1 (X1 , x) ×
W
π1 (X, x).
N
In the following two theorems, we reconstruct isomorphisms in Theorems 3.1 and 3.2 for n ≥ 2. In this case, since all groups are abelian, the direct product notation × must be replaced by the direct sum notation ⊕. Theorem 3.3. Let (X1 , x1 ) and (X2 , x2 ) be two pointed spaces, (X, x∗ ) = (X1 , x1 ) ∨ (X2 , x2 ), and n ≥ 2. If X is semilocally n-simply connected at x∗ , then Hn (X, x∗ ) ∼ = j∗ Hn (U1 ∨ U2 ) ⊕
πn (X, x∗ ),
N
for some neighbourhood U1 ∨ U2 of x∗ with the inclusion map j : U1 ∨ U2 → X. Theorem 3.4. Let (X1 , x1 ) and (X2 , x2 ) be two pointed spaces, (X, x∗ ) = (X1 , x1 ) ∨ (X2 , x2 ), and n ≥ 2. If πn (X1 , x) = e for some x1 = x ∈ X1 , then Hn (X, x) ∼ = Hn (X1 , x) ⊕
πn (X, x).
N
Ž. Virk [10] defined small 1-loop and studied small loop spaces. Note that a nullhomotopic loop is a small loop. H. Passandideh and F.H. Ghane [7] defined and studied the notions of n-homotopically Hausdorffness and small n-loops, for n ≥ 2. An n-loop α : (Sn , 1) → (X, x) is called small if it has a homotopic representative in every open neighbourhood of x. Theorem 3.5. Let (X1 , x1 ) and (X2 , x2 ) be two pointed spaces, (X, x∗ ) = (X1 , x1 ) ∨ (X2 , x2 ), and n ≥ 1. Also, let {x1 } be a closed subset in X1 , and x1 = x ∈ X1 . (i) All n-loops in X at x∗ are small if and only if Hn (X, x∗ ) = j∗ Hn (U1 ∨ U2 , x∗ ),
(5)
for any neighbourhoods U1 and U2 of x1 and x2 in X1 and X2 , respectively, when j : U1 ∨ U2 → X is the inclusion map. (ii) If all n-loops in X at x are small, then Hn (X, x) = Hn (X1 , x).
(6)
Proof. (i) Assume that all n-loops in X at x∗ are small. Let U1 and U2 be two arbitrary neighbourhoods W of x1 and x2 , respectively, and let [f ] ∈ N πn (X, x∗ ). We show that [f ] ∈ j∗ Hn (U1 ∨ U2 , x∗ ), and W then the equality (5) is obtained by Corollary 2.5. Since [f ] ∈ N πn (X, x∗ ), one can consider f as f |k≥K Snk = C|k≥K Snk for some K ∈ N. Moreover, since each n-loop in X is small at x∗ , any n-loop α is homotopic to some n-loop in U1 ∨ U2 , say α ˜ : Sn → U1 ∨ U2 , such that j ◦ α ˜ α. By induction on finite join of n-loops, f | k
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Fig. 2. The wedge sum of a circle and a cone on the Hawaiian earring.
Conversely, let α be an n-loop in X at x∗ . Consider the map f : (HEn , θ) → (X, x∗ ) so that f |Sn1 = α and f |Snk = c for k > 1. Then [f ] ∈ j∗ Hn (U1 ∨ U2 , x∗ ) by equality (5) for any neighbourhoods U1 and U2 . Let [f˜] be the element of Hn (U1 ∨ U2 , x∗ ) such that j ◦ f˜ = f . Hence, j ◦ f˜|Sn1 = f |Sn1 = α, that is α is homotopic to some n-loop in U1 ∨ U2 . Since U1 and U2 are arbitrary neighbourhoods, α is a small n-loop. (ii) Since X1 \ {x1 } is open and all n-loops at x in X are small, similar to the proof of the previous part, W one can show that N πn (X, x) ⊆ i∗ Hn (X1 \ {x1 }, x), where i : X1 \ {x1 } → X is the inclusion map. Moreover, i∗ Hn (X1 \ {x1 }, x) is contained in Hn (X1 , x) as a subgroup. Thus, by Corollary 2.5, the equality (6) holds. 2 Since a nullhomotopic n-loop is a special case of small n-loop, the equalities (5) and (6) hold for n-simply connected spaces. For example, Theorem 3.5 holds for two cones joining at their vertices which is not only n-simply connected, but also contractible. Recall that the 1st Griffiths space, the wedge sum of two cones, is not contractible, even more Griffiths [5] proved that it is not 1-simply connected. Example 3.6. For a space X, put Y = X×[−1,1] X×{0} the wedge sum of two cones over X at their vertices and x ˜t = [(x0 , t)] for t = 0. One can see that Y is contractible at the common point, and hence, it is n-simply H (X,x0 ) connected. By Theorem 3.5, Hn (Y, x ˜t ) = Hn (CX, x ˜t ). Also, by [1, Theorem 2.13], Hn (Y, x0 ) ∼ . = W n π (X,x ) k∈N
n
0
Let x∗ be the common vertex of the two cones. Then Hn (Y, x∗ ) is trivial, for Y is semilocally strongly W contractible at x∗ . By using Corollary 2.6, Hn (Y, x∗ ) = N πn (Y, x∗ ) which is trivial. The following example reveals that Theorem 3.5 does not hold, if some non small loop exists. Example 3.7. Let (X, x∗ ) = (C(HE1 ), θ) ∨ (S1 , 1) (see Fig. 2). If one assumes that H1 (X, x∗ ) = H1 (C(HE1 ), x∗ ), then the simple 1-loop in S1 must be nullhomotopic which is a contradiction. 4. Hawaiian groups of Griffiths spaces In this section, by generalizing the Griffiths space to higher dimensions and applying the results of Sections 2 and 3, we study the nth Hawaiian group of the nth Griffiths space, for n ≥ 1. Eda [3] introduced the free σ-product ×σN Z as the group consisting of all reduced σ-words, and then proved that it is isomorphic to π1 (HE1 , θ) [3, Theorem A.1]. To prove, Eda remarked that each 1-loop in the 1-dimensional Hawaiian earring is homotopic to some proper 1-loop [3, Lemma A.3]. A 1-loop α : ˙ → (HE1 , θ) is called proper whenever for each subinterval [a, b] ⊆ I, if α|[a,b] is nullhomotopic, then it (I, I) is constant. Also, O. Bogopolski and A. Zastrow proved that π1 (G1 , a) ∼ =
×σN Z ×σNe Z, ×σNo Z
N
,
(7)
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where No and Ne denote the set of odd and even numbers, respectively [2, Theorem 3.4]. This isomorphism is induced by the natural embedding ι : HE1 → G1 causing epimorphism π1 (ι) : ×σN Z → π1 (G1 , x∗ ) together N
with ×σNe Z, ×σNo Z as its kernel. The following lemma gives a useful description of the group H1 (HE1 , θ) which is used in sequel. Lemma 4.1. Let B be the subgroup of N ×σN Z consisting of all countably infinite tuples of reduced σ-words such that the number of components including letter of type m is finite, for all m ∈ N. Then H1 (HE1 , θ) ∼ = B. Proof. By [1, Theorem 2.9], group H1 (HE1 , θ) is isomorphic to the subgroup of N π1 (HE1 , θ) consisting of all sequences of homotopy classes of 1-loops with some representative converging uniformly to the constant 1-loop. We show that B equals this subgroup. Let {Ul ; l ∈ N} be the local basis at θ defined in the proof of [1, Theorem 2.9], and also let {[fk ]} ∈ 1 N π1 (HE , θ). Then {fk } converges uniformly to the constant 1-loop if and only if for each l ∈ N, there exists Kl ∈ N such that im(fk ) ⊆ Ul whenever k ≥ Kl . Assume that fk is the corresponding proper representative for all k ∈ N. The image of fk is contained in Ul if and only if im(fk ) ∩ S1m = {θ}, for all m < l, as for such m the intersection U ∩ S1m is contractible and fk has no trivial subpath. Therefore, {fk } converges uniformly to the constant 1-loop if and only if for each l ∈ N, there exists Kl ∈ N such that im(fk ) ∩ S1m = {θ} whenever k ≥ Kl . By [3, Theorem A.1], π1 (HE1 , θ) ∼ = ×σN Z, the group of reduced σ-words. Moreover, in a given reduced σ-word, the letter of type m exists if and only if the mth circle S1m of HE1 appears in the corresponding proper 1-loop. Therefore H1 (HE1 , θ) is isomorphic to the subgroup of N ×σN Z consisting of all countably infinite tuples of reduced σ-words such that the number of components including letter of type m is finite, for all m ∈ N. 2 The following theorem investigates the structure of the 1st Hawaiian group of the 1st Griffiths space at the common point, the two vertices and the other points. Let i : HE1 → G1 \ {v1 , v2 } be the embedding which maps (2m − 1)th circle onto the horizontal left mth circle and maps 2mth circle onto the horizontal right mth circle, for m ∈ N (see Fig. 1). Also, let j : G1 \ {v1 , v2 } → G1 be the inclusion map. Theorem 4.2. Let G1 be the 1st Griffiths space, x∗ the common point, a ∈ A ∪A , and x ∈ G1 \(A ∪A ∪{x∗ }). Then H1 (G1 , x∗ ) ∼ =
H1 (HE1 , θ) ∼ B , = N W σ σ Z i−1 ker j ∗ ∗ × Z, × Ne No N
W W H1 (HE1 , θ) B ×σN Z ∼ H1 (G1 , a) ∼ π (G , a) , × × = W = 1 1 W N 1 σ ×σNe Z, ×σNo Z N π1 (HE , θ) N ×N Z N N H1 (G1 , x) ∼ =
W N
×σN Z ×σNe Z, ×σNo Z
N
.
(i)
(ii)
(iii)
Proof. (i) Since all 1-loops at x∗ are small, Theorem 3.5 implies that H1 (G1 , x∗ ) = j∗ H1 (U1 ∨ U2 , x∗ ), when U1 and U2 are arbitrary neighbourhoods of x∗ in the two cones. Suppose that Um (m = 1, 2) is the whole of the corresponding cone except its vertex. Then U1 ∨ U2 = G1 \ {v1 , v2 } and thus H1 (G1 , x∗ ) = j∗ H1 (U1 ∨ U2 , x∗ ) = j∗ H1 (G1 \ {v1 , v2 }, x∗ ).
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Moreover, the embedded Hawaiian earring i(HE1 ) is a deformation retract of G1 \{v1 , v2 } with projection p : G1 \ {v1 , v2 } → HE1 as the retraction. Therefore, ∼ j∗ i∗ H1 (HE1 , θ). H1 (G1 , x∗ ) = j∗ H1 (G1 \ {v1 , v2 }, x∗ ) = By the First Isomorphism Theorem, i∗ H1 (HE1 , θ) ∼ H1 (HE1 , θ) j∗ i∗ H1 (HE1 , θ) ∼ . = = −1 ker j∗ i∗ ker j∗ W σ N σ In the following, we show that i−1 ∗ ker j∗ is mapped isomorphically onto group N ×Ne Z, ×No Z , by the same isomorphism which maps H1 (HE1 , θ) onto B, in Lemma 4.1. Let [g] ∈ i−1 ∗ ker j∗ . Then i∗ [g] ∈ ker j∗ , and therefore, j ◦i ◦g Cx∗ . Hence, the 1-loops (j ◦i ◦g|S1k )’s are nullhomotopic with some null convergent sequence of homotopies, say {Hk }. Thus, there exists K ∈ N such that imHk ⊆ G1 \ {v1 , v2 } for k ≥ K. Therefore, i ◦ g|S1k is nullhomotopic in G1 \ {v1 , v2 } for k ≥ K by null convergent homotopies {Hk }k≥K . Accordingly, i ◦ g|k≥K S1k is nullhomotpic in G \ {v1 , v2 }. Therefore, p ◦ i ◦ g|k≥K S1k = g|k≥K S1k is nullhomotopic in HE1 . Moreover, for k < K, j ◦ i ◦ g|Sk1 is nullhomotopic in G1 or equivalently [g|Sk1 ] ∈ ker π1 (ι). Note that N
ker π1 (ι) is mapped isomorphically onto ×σNe Z, ×σNo Z , by the same isomorphism mapping π1 (HE, θ) W N onto ×σN Z. Thus, [g] is corresponded injectively to an element of N ×σNe Z, ×σNo Z , where the correspondence is the same as the isomorphism mapping H1 (HE, θ) onto B, in Lemma 4.1. One can check that this correspondence is also surjective. (ii) By Theorem 3.2, H1 (G1 , a) ∼ = H1 (CHE1 , a) ×
W
π1 (G1 , a),
N
and by isomorphism (7), H1 (G1 , a) ∼ = H1 (CHE1 , a) ×
W
×σN Z N
N
×σNe Z, ×σNo Z
.
H (HE ,θ) Now by [1, Theorem 2.13] H1 (CHE1 , a) ∼ = W 1π1 (HE1 ,θ) . N W W By Lemma 4.1, H1 (HE1 , θ) ∼ = B, which maps subgroup N π1 (HE1 , θ) onto N ×σN Z. Consequently H1 (CHE1 , a) ∼ = WB×σ Z , and hence the isomorphism (ii) holds. N N W (iii) Obviously, G1 is semilocally strongly contractible at x. Therefore, H1 (G1 , x) ∼ = N π1 (G1 , x). 2 1
Eda and Kawamura [4] defined the nth Griffiths space by the wedge sum of two copies of cones on HEn at the origin for n ≥ 2. Furthermore, they proved that πn (Gn ) is trivial, for n ≥ 2 [4, Corollary 1.4]. In the following theorem we give some information on the structure of the nth Hawaiian group of the nth Griffiths space. Note that by Zk we mean the subgroup of N Z consisting of all countably infinite tuples with all zero components except possibly for any collection of the first k components, and thus Z1 Z2 Z3 · · · . Theorem 4.3. Let n ≥ 2, Gn be the nth Griffiths space, and a ∈ A ∪ A ∪ {x∗ }. Then Z Hn (HEn , θ) ∼ ∼ Hn (Gn , a) = = N N k , n N πn (HE , θ) k∈N NZ
(i)
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(ii) and Hn (Gn , x) is trivial, where x ∈ G1 \ (A ∪ A ∪ {x∗ }). Proof. (i) If a ∈ A ∪ A , then by Theorem 3.4, Hn (Gn , a) ∼ = Hn (CHEn , a) ⊕ N πn (Gn , a). Since πn (Gn , a) is trivial [4, Corollary 1.4], the second factor is deleted, and hence Hn (Gn , a) ∼ = H (CHEn , a). Using n n H (HE ,θ) n n n ∼ [1, Theorem 2.13], Hn (CHE , a) = πn (HEn ,θ) . Replacing Hn (HE , θ) with N N Z via [1, Theorem N 2.11], which maps N πn (HEn , θ) isomorphically onto the subgroup k∈N N Zk , the result holds. Let a = x∗ . By Corollary 2.5, Hn (Gn , x∗ ) = j∗ Hn (U1 ∨ U2 , x∗ ) N πn (Gn , x∗ ) for any neighbourhoods U1 and U2 of the origin in two cones, when j : U1 ∨ U2 → Gn is the inclusion. Since πn (Gn ) is trivial, Hn (Gn , x∗ ) = j∗ Hn (U1 ∨ U2 , x∗ ) for any neighbourhoods U1 and U2 . Put U1 to be the whole of one cone without its vertex, and U2 be the another cone without its vertex. Then U1 ∨ U2 = Gn \ {v, v }. Similar to the proof of Theorem 4.2, Part 1, one can replace j∗ Hn (U1 ∨ U2 , x∗ ) by j∗ i∗ Hn (HEn , θ) in the above isomorphism, where i : HEn → Gn \ {v, v } is the natural embedding. Hence Hn (Gn , x∗ ) ∼ = Hn (HEn ,θ) −1 . Similar to the proof of Theorem 4.2, Part 1, one can prove that i ker j = ker π (ι). ∗ n ∗ N i−1 ∗ ker j∗ Hn (HEn ,θ) n ∼ Since πn (Gn ) is trivial, ker πn (ι) = πn (HE , θ). Therefore, Hn (Gn , x∗ ) = . In the case n
N
πn (HE ,θ)
,θ) ∼ N N Z a ∈ A ∪ A , we show that Hnπ(HE = n k as required. N n (HE ,θ) k∈N N Z (ii) Since Gn is semilocally strongly contractible at x, Hn (Gn , x) ∼ = N πn (Gn ) by Corollary 2.6. Moreover, πn (Gn ) is trivial by [4, Corollary], and hence Hn (Gn , x) is trivial. 2 n
Acknowledgements The authors would like to thank the referee for his/her careful reading and useful comments and suggestions that helped to improve the article. This research was supported by a grant from Ferdowsi University of Mashhad-Graduate Studies (No. 42705). References [1] A. Babaee, B. Mashayekhy, H. Mirebrahimi, On Hawaiian groups of some topological spaces, Topol. Appl. 159 (8) (2012) 2043–2051. [2] O. Bogopolski, A. Zastrow, The word problem for some uncountable groups given by countable words, Topol. Appl. 159 (2012) 569–586. [3] K. Eda, Free σ products and noncommutatively slender groups, J. Algebra 148 (1992) 243–263. [4] K. Eda, K. Kawamura, Homotopy and homology groups of the n-dimensional Hawaiian earring, Fundam. Math. 165 (1) (2000) 17–28. [5] H.B. Griffiths, The fundamental group of two spaces with a common point, Q. J. Math., Oxford Ser. (2) 5 (1954) 175–190; H.B. Griffiths, correction: Q. J. Math., Oxford Ser. (2) 6 (1955) 154–155. [6] U.H. Karimov, D. Repovš, Hawaiian groups of topological spaces (Russian), Usp. Mat. Nauk 61 (5) (2006) 185–186; transl. in Russian: Russ. Math. Surv. 61 (5) (2006) 987–989. [7] H. Passandideh, F.H. Ghane, Homotopy properties of subsets of Euclidean spaces, Topol. Appl. 194 (2015) 202–211. [8] D.J.S. Robinson, A Course in the Group Theory, Springer-Verlag, New York, 1996. [9] R.M. Switzer, Algebraic Topology–Homotopy and Homology, Springer-Verlag, Berlin, 1975. [10] Ž. Virk, Small loop spaces, Topol. Appl. 157 (2010) 451–455.